We must return now to the nineteenth century. Presidential elections were not the only place where some strange developments were occurring. In the 1880s the chief clerk of the Census Bureau computed the apportionment of the House for all sizes from 275 to 350. After all, if adjusting the divisor can produce “interesting” results, might not the same happen with an adjustment of the House size? The clerk’s computations revealed what is now known as the “Alabama Paradox”: for a House size of 299, Alabama would receive 8 Representatives, but if the House increased to 300, Alabama’s delegation would have dropped to 7. It would seem that any apportionment method exhibiting such undesirable behavior might be excluded from further consideration, although that is not the case. In fact, some today argue that Hamilton’s method is superior to all of the others, despite its paradoxical behavior.

See the spreadsheet Alabama paradox for an example of the effect using the census of 1870 and changing the House size from 270 to 280. In this illustration, it is Rhode Island which suffers the indignity of dropping from 2 Representatives in a 270 member House to 1 in a House of 280.

In the 1880s and 1890s, House sizes were chosen so that the calculations based on Hamilton’s method and Webster’s method would agree: 325 seats in the 1880s and 356 in the 1890s. But after the census of 1900, new “numbers games” surfaced.

Apportionments based on Hamilton’s method were calculated for all House sizes from 350 through 400. From sizes 382 through 391, Maine’s delegation switched from 3 to 4 seats and back again five different times. Colorado only suffered Maine’s plight once: it was to receive 3 seats for all values computed except for a House size of 357, in which case it would receive 2. The Chair of the House Select Committee on the 12th Census, no friend of the Colorado Populists, proposed a House size of 357. His proposal, alas, was defeated.

In fact, so was Hamilton’s method, once and for all. At least twenty years after the Alabama paradox had come to light, and after repeated observations of the numerical fluctuations that afflicted various states, Congress voted to replace Hamilton with Webster. The House size was set at 386, so that, of course, no state would lose a seat. It repeated itself ten years later when Webster was re-authorized with a House size of 433 (again, the smallest size so that no state loses a seat).

Before we move too fast into the twentieth century, let us return to 1907 and the entry of Oklahoma into the Union. Although Congress had adopted Webster, clerks for the Representatives, and others, examined what would have happened had Hamilton still been in place. They discovered yet another flaw in the method, known as the “New States Paradox.” In 1907 the House size was 386. According to its population at the time, upon entry into the Union Oklahoma was entitled to 5 Representatives, bringing the House size to 391. But when the Hamilton apportionment was re-calculated based on the new House size of 391, New York lost a seat to Maine! See the spreadsheet Hamilton new states paradox for an illustration of what happened. In brief, the new House size caused a shift in the remainders that Hamilton uses to apportion the surplus Representatives after all state quotas have been rounded down.